Symplectic integrators for classical spin systems

Steinigeweg, Robin; Schmidt, Heinz‐Jürgen

doi:10.1016/j.cpc.2005.12.023

Cited by 18 publications

(16 citation statements)

References 22 publications

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“…Unfortunately, q = π/20 is firstly realized for a system which consists of N = 40 layers and such a system already is too large for the application of numerically exact diagonalization. However, approximative numerical integrators may be applied, e.g., on the basis of a Suzuki-Trotter decomposition of the time evolution operator [27,28,29]. In detail we choose a pure initial state |ψ q (0) and apply a fourth order Suzuki-Trotter integrator in order to obtain the time evolution |ψ q (t) of this initial state and to evaluate the actual expectation value p q (t) = ψ q (t)|p q |ψ q (t) .…”

Section: Numerical Verificationmentioning

confidence: 99%

Transport in the three-dimensional Anderson model: an analysis of the dynamics at scales below the localization length

2010

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Single-particle transport in disordered potentials is investigated on scales below the localization length. The dynamics on those scales is concretely analyzed for the 3-dimensional Anderson model with Gaussian on-site disorder. This analysis particularly includes the dependence of characteristic transport quantities on the amount of disorder and the energy interval, e.g., the mean free path which separates ballistic and diffusive transport regimes. For these regimes mean velocities, respectively diffusion constants are quantitatively given. By the use of the Boltzmann equation in the limit of weak disorder we reveal the known energy-dependencies of transport quantities. By an application of the time-convolutionless (TCL) projection operator technique in the limit of strong disorder we find evidence for much less pronounced energy dependencies. All our results are partially confirmed by the numerically exact solution of the time-dependent Schrödinger equation or by approximative numerical integrators. A comparison with other findings in the literature is additionally provided.

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Section: Numerical Verificationmentioning

confidence: 99%

Transport in the three-dimensional Anderson model: an analysis of the dynamics at scales below the localization length

2010

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“…2). Note that the integration in this case already requires approximative numerical integrators like, e.g., Suzuki-Trotter decompositions [13]. A numerical integration of systems with larger N rapidly becomes unfeasible but an analysis based on Eq.…”

Section: Psfrag Replacementsmentioning

confidence: 99%

Length scale dependent diffusion in the Anderson model at high temperatures

Steinigeweg

Gemmer

2010

Physica E: Low-dimensional Systems and Nanostructures

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We investigate a single particle on a 3-dimensional, cubic lattice with a random on-site potential (3D Anderson model). We concretely address the question whether or not the dynamics of the particle is in full accord with the diffusion equation. Our approach is based on the time-convolutionless (TCL) projection operator technique and allows for a detailed investigation of this question at high temperatures. It turns out that diffusive dynamics is to be expected for a rather short range of wavelengths, even if the amount of disorder is tuned to maximize this range. Our results are partially counterchecked by the numerical solution of the full time-dependent Schrödinger equation.

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“…The overall conservation of the local densities finally leads to α µ,µ−1 (t) = −α µ,µ (t)/2. Applying the Fourier transform to (25) yieldṡ…”

Section: Extension Of the Projection And Fourth Order Tclmentioning

confidence: 99%

Projection operator approach to transport in complex single-particle quantum systems

et al. 2009

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We discuss the time-convolutionless (TCL) projection operator approach to transport in closed quantum systems. The projection onto local densities of quantities such as energy, magnetization, particle number, etc. yields the reduced dynamics of the respective quantities in terms of a systematic perturbation expansion. In particular, the lowest order contribution of this expansion is used as a strategy for the analysis of transport in "modular" quantum systems corresponding to quasi one-dimensional structures which consist of identical or similar many-level subunits. Such modular quantum systems are demonstrated to represent many physical situations and several examples of complex single-particle models are analyzed in detail. For these quantum systems lowest order TCL is shown to represent an efficient tool which also allows to investigate the dependence of transport on the considered length scale. To estimate the range of validity of the obtained equations of motion we extend the standard projection to include additional degrees of freedom which model non-Markovian effects of higher orders. PACS. 05.60.Gg Quantum transport -05.30.-d Quantum statistical mechanics -05.70.Ln Nonequilibrium and irreversible thermodynamics

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Symplectic integrators for classical spin systems

Cited by 18 publications

References 22 publications

Transport in the three-dimensional Anderson model: an analysis of the dynamics at scales below the localization length

Transport in the three-dimensional Anderson model: an analysis of the dynamics at scales below the localization length

Length scale dependent diffusion in the Anderson model at high temperatures

Projection operator approach to transport in complex single-particle quantum systems

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